napier's logarithms : the development of his theory. 133 



as 10 7 . It may be that Biirgi chose 10 8 in his tables for a 

 similar reason. With our notation Napier's sines would 

 correspond to 7-Figure Tables of Natural Sines, etc. If 

 greater accuracy were required, the radius was taken as 

 10 10 , and sometimes even a higlier power of 10 was used. 

 These sines, etc., following Glaisher, 1 we shall refer to as 

 line-sines, etc. 



The Second Stage. 



§ 3. Napier opened out entirely fresh ground, when he 

 passed to his kinematical definition of the logarithm of a 

 sine or number. By this definition he associated with the 

 sine, as it continually diminished from 10 7 for 90° to zero 

 for 0°, a number which he called its logarithm ; and the 

 logarithm continually increased from 0, for the sine of 90°, 

 to infinity, for the sine of 0°. 



The fundamental proposition in Napier's theory in the 

 Descriptio (1614) and the Constructio (1619) is to be found 

 in Prop. I of the Descriptio: 



The logarithmes of proportionall numbers and quantities 

 are equally differing. 2 



And in Section 36 of the Constructio it appears as the 

 logari thins of similarly proportioned sines are equidifferent. 



Glaisher has introduced a convenient notation nl r x for 

 Napier's logarithm, in this system, when the radius is 10*. 

 He also uses Sin r x for the line-sine of the angle x, when 

 the radius is 10 r , and he keeps the symbol sin oc for the sine 

 in the modern sense of the term. With this notation we 

 have sin x = Sinrx 



W 



1 Quarterly Journal, Vol. 46, p. 125 (1916). To this paper I am indebted, 

 not only for a most convenient notation for the different systems of 

 logarithms, but also for an account of SpeidelFs work, hitherto inaccess- 

 ible to me. 



2 In quoting the Descriptio I follow Wright's version, and for the Con- 

 structio I adopt Macdonald's. 



